@article{Roos2019, author = {Roos, Saskia}, title = {The Dirac operator under collapse to a smooth limit space}, series = {Annals of global analysis and geometry}, volume = {57}, journal = {Annals of global analysis and geometry}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {0232-704X}, doi = {10.1007/s10455-019-09691-8}, pages = {121 -- 151}, year = {2019}, abstract = {Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B.}, language = {en} } @article{GueneysuKeller2018, author = {G{\"u}neysu, Batu and Keller, Matthias}, title = {Scattering the Geometry of Weighted Graphs}, series = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, volume = {21}, journal = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, number = {3}, publisher = {Springer}, address = {Dordrecht}, issn = {1385-0172}, doi = {10.1007/s11040-018-9285-1}, pages = {15}, year = {2018}, abstract = {Given two weighted graphs (X, b(k), m(k)), k = 1, 2 with b(1) similar to b(2) and m(1) similar to m(2), we prove a weighted L-1-criterion for the existence and completeness of the wave operators W-+/- (H-2, H-1, I-1,I-2), where H-k denotes the natural Laplacian in l(2)(X, m(k)) w.r.t. (X, b(k), m(k)) and I-1,I-2 the trivial identification of l(2)(X, m(1)) with l(2) (X, m(2)). In particular, this entails a general criterion for the absolutely continuous spectra of H-1 and H-2 to be equal.}, language = {en} } @article{FedchenkoTarkhanov2017, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {A Rado theorem for the porous medium equation}, series = {Boletin de la Sociedad Matem{\´a}tica Mexicana}, volume = {24}, journal = {Boletin de la Sociedad Matem{\´a}tica Mexicana}, number = {2}, publisher = {Springer}, address = {Cham}, issn = {1405-213X}, doi = {10.1007/s40590-017-0169-3}, pages = {427 -- 437}, year = {2017}, abstract = {We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi.}, language = {en} } @article{DimitrovaKoppitz2017, author = {Dimitrova, Ilinka and Koppitz, J{\"o}rg}, title = {On the semigroup of all partial fence-preserving injections on a finite set}, series = {Journal of Algebra and Its Applications}, volume = {16}, journal = {Journal of Algebra and Its Applications}, number = {12}, publisher = {World Scientific}, address = {Singapore}, issn = {0219-4988}, doi = {10.1142/S0219498817502231}, pages = {14}, year = {2017}, abstract = {For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn; infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.}, language = {en} } @article{PerkinsPernaAdrianetal.2019, author = {Perkins, Daniel M. and Perna, Andrea and Adrian, Rita and Cermeno, Pedro and Gaedke, Ursula and Huete-Ortega, Maria and White, Ethan P. and Yvon-Durocher, Gabriel}, title = {Energetic equivalence underpins the size structure of tree and phytoplankton communities}, series = {Nature Communications}, volume = {10}, journal = {Nature Communications}, publisher = {Nature Publ. Group}, address = {London}, issn = {2041-1723}, doi = {10.1038/s41467-018-08039-3}, pages = {8}, year = {2019}, abstract = {The size structure of autotroph communities - the relative abundance of small vs. large individuals - shapes the functioning of ecosystems. Whether common mechanisms underpin the size structure of unicellular and multicellular autotrophs is, however, unknown. Using a global data compilation, we show that individual body masses in tree and phytoplankton communities follow power-law distributions and that the average exponents of these individual size distributions (ISD) differ. Phytoplankton communities are characterized by an average ISD exponent consistent with three-quarter-power scaling of metabolism with body mass and equivalence in energy use among mass classes. Tree communities deviate from this pattern in a manner consistent with equivalence in energy use among diameter size classes. Our findings suggest that whilst universal metabolic constraints ultimately underlie the emergent size structure of autotroph communities, divergent aspects of body size (volumetric vs. linear dimensions) shape the ecological outcome of metabolic scaling in forest vs. pelagic ecosystems.}, language = {en} } @article{LeungLeutbecherReichetal.2019, author = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.}, title = {Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier}, series = {Journal of the atmospheric sciences}, volume = {76}, journal = {Journal of the atmospheric sciences}, number = {12}, publisher = {American Meteorological Soc.}, address = {Boston}, issn = {0022-4928}, doi = {10.1175/JAS-D-19-0057.1}, pages = {3883 -- 3892}, year = {2019}, abstract = {The accepted idea that there exists an inherent finite-time barrier in deterministically predicting atmospheric flows originates from Edward N. Lorenz's 1969 work based on two-dimensional (2D) turbulence. Yet, known analytic results on the 2D Navier-Stokes (N-S) equations suggest that one can skillfully predict the 2D N-S system indefinitely far ahead should the initial-condition error become sufficiently small, thereby presenting a potential conflict with Lorenz's theory. Aided by numerical simulations, the present work reexamines Lorenz's model and reviews both sides of the argument, paying particular attention to the roles played by the slope of the kinetic energy spectrum. It is found that when this slope is shallower than -3, the Lipschitz continuity of analytic solutions (with respect to initial conditions) breaks down as the model resolution increases, unless the viscous range of the real system is resolved—which remains practically impossible. This breakdown leads to the inherent finite-time limit. If, on the other hand, the spectral slope is steeper than -3, then the breakdown does not occur. In this way, the apparent contradiction between the analytic results and Lorenz's theory is reconciled.}, language = {en} } @misc{BeckusBellissardDeNittis2019, author = {Beckus, Siegfried and Bellissard, Jean and De Nittis, Giuseppe}, title = {Corrigendum to: Spectral continuity for aperiodic quantum systems I. General theory. - [Journal of functional analysis. - 275 (2018), 11, S. 2917 - 2977]}, series = {Journal of functional analysis}, volume = {277}, journal = {Journal of functional analysis}, number = {9}, publisher = {Elsevier}, address = {San Diego}, issn = {0022-1236}, doi = {10.1016/j.jfa.2019.06.001}, pages = {3351 -- 3353}, year = {2019}, abstract = {A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results.}, language = {en} } @article{FernandesKoppitzMusunthia2019, author = {Fernandes, Vitor H. and Koppitz, J{\"o}rg and Musunthia, Tiwadee}, title = {The Rank of the Semigroup of All Order-Preserving Transformations on a Finite Fence}, series = {Bulletin of the Malaysian Mathematical Sciences Society volume}, volume = {42}, journal = {Bulletin of the Malaysian Mathematical Sciences Society volume}, number = {5}, publisher = {Malaysian mathematical sciences sciences soc}, address = {Pulau Punang}, issn = {0126-6705}, doi = {10.1007/s40840-017-0598-1}, pages = {2191 -- 2211}, year = {2019}, abstract = {A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup TFn of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of TFn and provide a minimal generating set for TFn. Moreover, a formula for the number of idempotents in TFn is given.}, language = {en} } @article{ShcherbakovZhuangZoelleretal.2019, author = {Shcherbakov, Robert and Zhuang, Jiancang and Z{\"o}ller, Gert and Ogata, Yosihiko}, title = {Forecasting the magnitude of the largest expected earthquake}, series = {Nature Communications}, volume = {10}, journal = {Nature Communications}, publisher = {Nature Publishing Group}, address = {London}, issn = {2041-1723}, doi = {10.1038/s41467-019-11958-4}, pages = {11}, year = {2019}, abstract = {The majority of earthquakes occur unexpectedly and can trigger subsequent sequences of events that can culminate in more powerful earthquakes. This self-exciting nature of seismicity generates complex clustering of earthquakes in space and time. Therefore, the problem of constraining the magnitude of the largest expected earthquake during a future time interval is of critical importance in mitigating earthquake hazard. We address this problem by developing a methodology to compute the probabilities for such extreme earthquakes to be above certain magnitudes. We combine the Bayesian methods with the extreme value theory and assume that the occurrence of earthquakes can be described by the Epidemic Type Aftershock Sequence process. We analyze in detail the application of this methodology to the 2016 Kumamoto, Japan, earthquake sequence. We are able to estimate retrospectively the probabilities of having large subsequent earthquakes during several stages of the evolution of this sequence.}, language = {en} } @article{KhalilSchulze2019, author = {Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Calculus on a Manifold with Edge and Boundary}, series = {Complex analysis and operator theory}, volume = {13}, journal = {Complex analysis and operator theory}, number = {6}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-018-0800-y}, pages = {2627 -- 2670}, year = {2019}, abstract = {We study elements of the calculus of boundary value problems in a variant of Boutet de Monvel's algebra (Acta Math 126:11-51, 1971) on a manifold N with edge and boundary. If the boundary is empty then the approach corresponds to Schulze (Symposium on partial differential equations (Holzhau, 1988), BSB Teubner, Leipzig, 1989) and other papers from the subsequent development. For non-trivial boundary we study Mellin-edge quantizations and compositions within the structure in terms a new Mellin-edge quantization, compared with a more traditional technique. Similar structures in the closed case have been studied in Gil et al.}, language = {en} } @article{ConfortiKosenkovaRoelly2019, author = {Conforti, Giovanni and Kosenkova, Tetiana and Roelly, Sylvie}, title = {Conditioned Point Processes with Application to Levy Bridges}, series = {Journal of theoretical probability}, volume = {32}, journal = {Journal of theoretical probability}, number = {4}, publisher = {Springer}, address = {New York}, issn = {0894-9840}, doi = {10.1007/s10959-018-0863-8}, pages = {2111 -- 2134}, year = {2019}, abstract = {Our first result concerns a characterization by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalized version of Mecke's formula. En passant, it also allows us to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump L{\´e}vy process in Rd with a height a can be interpreted as a Poisson point process on space-time conditioned by pinning its first moment to a, our approach allows us to characterize bridges of L{\´e}vy processes by means of a functional equation. The latter result has two direct applications: First, we obtain a constructive and simple way to sample L{\´e}vy bridge dynamics; second, it allows us to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed L{\´e}vy processes like periodic Ornstein-Uhlenbeck processes driven by L{\´e}vy noise.}, language = {en} } @article{StaniforthWoodReich2006, author = {Staniforth, Andrew and Wood, Nigel and Reich, Sebastian}, title = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations}, series = {Quarterly journal of the Royal Meteorological Society}, volume = {132}, journal = {Quarterly journal of the Royal Meteorological Society}, number = {621C}, publisher = {Wiley}, address = {Weinheim}, issn = {0035-9009}, doi = {10.1256/qj.06.30}, pages = {3107 -- 3116}, year = {2006}, abstract = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes.}, language = {en} } @article{KirscheBoeckmann2006, author = {Kirsche, Andreas and B{\"o}ckmann, Christine}, title = {Pade iteration method for regularization}, series = {Applied mathematics and computation}, volume = {180}, journal = {Applied mathematics and computation}, number = {2}, publisher = {Elsevier}, address = {New York}, issn = {0096-3003}, doi = {10.1016/j.amc.2006.01.011}, pages = {648 -- 663}, year = {2006}, abstract = {In this study we present iterative regularization methods using rational approximations, in particular, Pade approximants, which work well for ill-posed problems. We prove that the (k,j)-Pade method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Pade method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Bockmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385-397] where this method was treated to solve ill-conditioned equation systems. (c) 2006 Elsevier Inc. All rights reserved.}, language = {en} } @article{Reich2006, author = {Reich, Sebastian}, title = {Linearly implicit time stepping methods for numerical weather prediction}, series = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians}, volume = {46}, journal = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians}, publisher = {Springer}, address = {Dordrecht}, issn = {0006-3835}, doi = {10.1007/s10543-006-0065-0}, pages = {607 -- 616}, year = {2006}, abstract = {The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.}, language = {en} } @article{BaerStrohmaier2019, author = {B{\"a}r, Christian and Strohmaier, Alexander}, title = {An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary}, series = {American Journal of Mathematics}, volume = {141}, journal = {American Journal of Mathematics}, number = {5}, publisher = {Johns Hopkins Univ. Press}, address = {Baltimore}, issn = {0002-9327}, doi = {10.1353/ajm.2019.0037}, pages = {1421 -- 1455}, year = {2019}, abstract = {We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.}, language = {en} } @article{Lewandowski2022, author = {Lewandowski, Max}, title = {Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes}, series = {Journal of mathematical physics}, volume = {63}, journal = {Journal of mathematical physics}, number = {1}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/5.0055753}, pages = {34}, year = {2022}, abstract = {According to Radzikowski's celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF-G˜F+G˜A-G˜R) in the sense of Duistermaat and H{\"o}rmander [Acta Math. 128, 183-269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by B{\"a}r, Ginoux, and Pf{\"a}ffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Z{\"u}rich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF-G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.}, language = {en} } @article{DeneckeKoppitzŠtrakov2006, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Štrakov, Slavčo}, title = {Multi-hypersubstitutions and colored solid varieties}, series = {International journal of algebra and computation}, volume = {16}, journal = {International journal of algebra and computation}, number = {4}, publisher = {World Scient. Publ.}, address = {Singapore}, issn = {0218-1967}, doi = {10.1142/S0218196706003189}, pages = {797 -- 815}, year = {2006}, abstract = {Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multihypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.}, language = {en} } @article{Denecke2019, author = {Denecke, Klaus-Dieter}, title = {The partial clone of linear formulas}, series = {Siberian mathematical journal}, volume = {60}, journal = {Siberian mathematical journal}, number = {4}, publisher = {Pleiades Publ.}, address = {New York}, issn = {0037-4466}, doi = {10.1134/S0037446619040037}, pages = {572 -- 584}, year = {2019}, abstract = {A term t is linear if no variable occurs more than once in t. An identity s ≈ t is said to be linear if s and t are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra Formclonelin(τ, τ′). This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type τ to the linear hypersubstitutions of type (τ, τ′) for algebraic systems. The extensions of linear hypersubstitutions of type (τ, τ′) send linear formulas to linear formulas, presenting weak endomorphisms of Formclonelin(τ, τ′).}, language = {en} } @article{LekkoksungDenecke2019, author = {Lekkoksung, Nareupanat and Denecke, Klaus-Dieter}, title = {The partial clone of linear tree languages}, series = {Siberian mathematical journal}, volume = {60}, journal = {Siberian mathematical journal}, number = {3}, publisher = {Pleiades Publ.}, address = {New York}, issn = {0037-4466}, doi = {10.1134/S0037446619030121}, pages = {497 -- 507}, year = {2019}, abstract = {A term, also called a tree, is said to be linear, if each variable occurs in the term only once. The linear terms and sets of linear terms, the so-called linear tree languages, play some role in automata theory and in the theory of formal languages in connection with recognizability. We define a partial superposition operation on sets of linear trees of a given type and study the properties of some many-sorted partial clones that have sets of linear trees as elements and partial superposition operations as fundamental operations. The endomorphisms of those algebras correspond to nondeterministic linear hypersubstitutions.}, language = {en} } @phdthesis{Supaporn2014, author = {Supaporn, Worakrit}, title = {Categorical equivalence of clones}, pages = {89}, year = {2014}, language = {en} } @phdthesis{Trappmann2007, author = {Trappmann, Henryk}, title = {Arborescent numbers : higher arithmetic operations and division trees}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15247}, school = {Universit{\"a}t Potsdam}, year = {2007}, abstract = {The overall program "arborescent numbers" is to similarly perform the constructions from the natural numbers (N) to the positive fractional numbers (Q+) to positive real numbers (R+) beginning with (specific) binary trees instead of natural numbers. N can be regarded as the associative binary trees. The binary trees B and the left-commutative binary trees P allow the hassle-free definition of arbitrary high arithmetic operations (hyper ... hyperpowers). To construct the division trees the algebraic structure "coppice" is introduced which is a group with an addition over which the multiplication is right-distributive. Q+ is the initial associative coppice. The present work accomplishes one step in the program "arborescent numbers". That is the construction of the arborescent equivalent(s) of the positive fractional numbers. These equivalents are the "division binary trees" and the "fractional trees". A representation with decidable word problem for each of them is given. The set of functions f:R1->R1 generated from identity by taking powers is isomorphic to P and can be embedded into a coppice by taking inverses.}, language = {en} } @article{FischerKeller2021, author = {Fischer, Florian and Keller, Matthias}, title = {Riesz decompositions for Schr{\"o}dinger operators on graphs}, series = {Journal of mathematical analysis and applications}, volume = {495}, journal = {Journal of mathematical analysis and applications}, number = {1}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2020.124674}, pages = {22}, year = {2021}, abstract = {We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.}, language = {en} } @article{Denecke2020, author = {Denecke, Klaus-Dieter}, title = {Partial clones}, series = {Asian-European journal of mathematics : AEJM}, volume = {13}, journal = {Asian-European journal of mathematics : AEJM}, number = {8}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557120501612}, pages = {19}, year = {2020}, abstract = {A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.}, language = {en} } @article{Baer2021, author = {B{\"a}r, Christian}, title = {The Faddeev-LeVerrier algorithm and the Pfaffian}, series = {Linear algebra and its applications}, volume = {630}, journal = {Linear algebra and its applications}, publisher = {Elsevier}, address = {New York}, issn = {0024-3795}, doi = {10.1016/j.laa.2021.07.023}, pages = {39 -- 55}, year = {2021}, abstract = {We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.}, language = {en} } @misc{KleinRosenberger2018, author = {Klein, Markus and Rosenberger, Elke}, title = {The tunneling effect for a class of difference operators}, series = {Reviews in Mathematical Physics}, volume = {30}, journal = {Reviews in Mathematical Physics}, number = {4}, publisher = {World Scientific}, address = {Singapore}, issn = {0129-055X}, doi = {10.1142/S0129055X18300029}, pages = {42}, year = {2018}, abstract = {We analyze a general class of self-adjoint difference operators H-epsilon = T-epsilon + V-epsilon on l(2)((epsilon Z)(d)), where V-epsilon is a multi-well potential and v(epsilon) is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30-35]). Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs. The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian H-epsilon is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance d induced by H-epsilon, we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first n eigenvalues of H-epsilon converge to the first n eigenvalues of the direct sum of harmonic oscillators on R-d located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of H-epsilon. These are obtained from eigenfunctions or quasimodes for the operator H-epsilon acting on L-2(R-d), via restriction to the lattice (epsilon Z)(d). Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrodinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted l(2)-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two "wells" (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrodinger operator in [22].}, language = {en} } @misc{ZoellerHolschneider2018, author = {Z{\"o}ller, Gert and Holschneider, Matthias}, title = {Reply to "Comment on 'The Maximum Possible and the Maximum Expected Earthquake Magnitude for Production-Induced Earthquakes at the Gas Field in Groningen, The Netherlands' by Gert Z{\"o}ller and Matthias Holschneider" by Mathias Raschke}, series = {Bulletin of the Seismological Society of America}, volume = {108}, journal = {Bulletin of the Seismological Society of America}, number = {2}, publisher = {Seismological Society of America}, address = {Albany}, issn = {0037-1106}, doi = {10.1785/0120170131}, pages = {1029 -- 1030}, year = {2018}, language = {en} } @article{Gerlach2018, author = {Gerlach, Moritz Reinhardt}, title = {Convergence of dynamics and the Perron-Frobenius operator}, series = {Israel Journal of Mathematics}, volume = {225}, journal = {Israel Journal of Mathematics}, number = {1}, publisher = {Hebrew univ magnes press}, address = {Jerusalem}, issn = {0021-2172}, doi = {10.1007/s11856-018-1671-7}, pages = {451 -- 463}, year = {2018}, abstract = {We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system.}, language = {en} } @article{LiuMuenchPeyerimhoff2018, author = {Liu, Shiping and M{\"u}nch, Florentin and Peyerimhoff, Norbert}, title = {Bakry-Emery curvature and diameter bounds on graphs}, series = {Calculus of variations and partial differential equations}, volume = {57}, journal = {Calculus of variations and partial differential equations}, number = {2}, publisher = {Springer}, address = {Heidelberg}, issn = {0944-2669}, doi = {10.1007/s00526-018-1334-x}, pages = {9}, year = {2018}, abstract = {We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry-{\´E}mery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672-698, 2018) and Horn et al. (J f{\"u}r die reine und angewandte Mathematik (Crelle's J), 2017, https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry-{\´E}mery curvature functions of graphs, 2016).}, language = {en} } @article{KellerSchwarz2018, author = {Keller, Matthias and Schwarz, Michael}, title = {The Kazdan-Warner equation on canonically compactifiable graphs}, series = {Calculus of variations and partial differential equations}, volume = {57}, journal = {Calculus of variations and partial differential equations}, number = {2}, publisher = {Springer}, address = {Heidelberg}, issn = {0944-2669}, doi = {10.1007/s00526-018-1329-7}, pages = {18}, year = {2018}, abstract = {We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.}, language = {en} } @article{LesurWardinskiBaerenzungetal.2017, author = {Lesur, Vincent and Wardinski, Ingo and B{\"a}renzung, Julien and Holschneider, Matthias}, title = {On the frequency spectra of the core magnetic field Gauss coefficients}, series = {Physics of the earth and planetary interiors}, volume = {276}, journal = {Physics of the earth and planetary interiors}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0031-9201}, doi = {10.1016/j.pepi.2017.05.017}, pages = {145 -- 158}, year = {2017}, abstract = {From monthly mean observatory data spanning 1957-2014, geomagnetic field secular variation values were calculated by annual differences. Estimates of the spherical harmonic Gauss coefficients of the core field secular variation were then derived by applying a correlation based modelling. Finally, a Fourier transform was applied to the time series of the Gauss coefficients. This process led to reliable temporal spectra of the Gauss coefficients up to spherical harmonic degree 5 or 6, and down to periods as short as 1 or 2 years depending on the coefficient. We observed that a k(-2) slope, where k is the frequency, is an acceptable approximation for these spectra, with possibly an exception for the dipole field. The monthly estimates of the core field secular variation at the observatory sites also show that large and rapid variations of the latter happen. This is an indication that geomagnetic jerks are frequent phenomena and that significant secular variation signals at short time scales - i.e. less than 2 years, could still be extracted from data to reveal an unexplored part of the core dynamics.}, language = {en} } @article{GerlachGlueck2017, author = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen}, title = {On a convergence theorem for semigroups of positive integral operators}, series = {Comptes Rendus Mathematique}, volume = {355}, journal = {Comptes Rendus Mathematique}, publisher = {Elsevier}, address = {Paris}, issn = {1631-073X}, doi = {10.1016/j.crma.2017.07.017}, pages = {973 -- 976}, year = {2017}, abstract = {We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive -semigroup on an -space is strongly convergent in case it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.}, language = {en} } @article{BrungsGraeter2017, author = {Brungs, Hans H. and Gr{\"a}ter, Joachim}, title = {On central extensions of SL(2, F) admitting left-orderings}, series = {Journal of Algebra}, volume = {486}, journal = {Journal of Algebra}, publisher = {Elsevier}, address = {San Diego}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2017.05.025}, pages = {288 -- 327}, year = {2017}, abstract = {For an arbitrary euclidean field F we introduce a central extension (G(F), Phi) of SL(2, F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order preserving bijections of the convex hull of Q in F. If F = R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2, R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.}, language = {en} } @article{SomogyvariReich2020, author = {Somogyv{\´a}ri, M{\´a}rk and Reich, Sebastian}, title = {Convergence tests for transdimensional Markov chains in geoscience imaging}, series = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences}, volume = {52}, journal = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences}, number = {5}, publisher = {Springer}, address = {Heidelberg}, issn = {1874-8961}, doi = {10.1007/s11004-019-09811-x}, pages = {651 -- 668}, year = {2020}, abstract = {Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.}, language = {en} } @article{Ly2020, author = {Ly, Ibrahim}, title = {A Cauchy problem for the Cauchy-Riemann operator}, series = {Afrika Matematika}, volume = {32}, journal = {Afrika Matematika}, number = {1-2}, publisher = {Springer}, address = {Heidelberg}, issn = {1012-9405}, doi = {10.1007/s13370-020-00810-4}, pages = {69 -- 76}, year = {2020}, abstract = {We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.}, language = {en} } @unpublished{LyTarkhanov2013, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Generalised Beltrami equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67416}, year = {2013}, abstract = {We enlarge the class of Beltrami equations by developping a stability theory for the sheaf of solutions of an overdetermined elliptic system of first order homogeneous partial differential equations with constant coefficients in the Euclidean space.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {A Rad{\´o} theorem for p-harmonic functions}, volume = {4}, number = {3}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-71492}, pages = {10}, year = {2015}, abstract = {Let A be a nonlinear differential operator on an open set X in R^n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A (u) = 0 in the complement of S of class F satisfies this equation weakly in all of X. For the most extensively studied classes F we show conditions on S which guarantee that S is removable for F relative to A.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Asymptotic expansions at nonsymmetric cuspidal points}, volume = {4}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78199}, pages = {11}, year = {2015}, abstract = {We study asymptotics of solutions to the Dirichlet problem in a domain whose boundary contains a nonsymmetric conical point. We establish a complete asymptotic expansion of solutions near the singular point.}, language = {en} } @article{MalassTarkhanov2020, author = {Malass, Ihsane and Tarkhanov, Nikolaj Nikolaevič}, title = {A perturbation of the de Rham complex}, series = {Journal of Siberian Federal University : Mathematics \& Physics}, volume = {13}, journal = {Journal of Siberian Federal University : Mathematics \& Physics}, number = {5}, publisher = {Siberian Federal University}, address = {Krasnojarsk}, issn = {1997-1397}, doi = {10.17516/1997-1397-2020-13-5-519-532}, pages = {519 -- 532}, year = {2020}, abstract = {We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.}, language = {en} } @phdthesis{Jakobs2019, author = {Jakobs, Friedrich}, title = {Dubrovin-rings and their connection to Hughes-free skew fields of fractions}, doi = {10.25932/publishup-43556}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-435561}, school = {Universit{\"a}t Potsdam}, pages = {ix, 62}, year = {2019}, abstract = {One method of embedding groups into skew fields was introduced by A. I. Mal'tsev and B. H. Neumann (cf. [18, 19]). If G is an ordered group and F is a skew field, the set F((G)) of formal power series over F in G with well-ordered support forms a skew field into which the group ring F[G] can be embedded. Unfortunately it is not suficient that G is left-ordered since F((G)) is only an F-vector space in this case as there is no natural way to define a multiplication on F((G)). One way to extend the original idea onto left-ordered groups is to examine the endomorphism ring of F((G)) as explored by N. I. Dubrovin (cf. [5, 6]). It is possible to embed any crossed product ring F[G; η, σ] into the endomorphism ring of F((G)) such that each non-zero element of F[G; η, σ] defines an automorphism of F((G)) (cf. [5, 10]). Thus, the rational closure of F[G; η, σ] in the endomorphism ring of F((G)), which we will call the Dubrovin-ring of F[G; η, σ], is a potential candidate for a skew field of fractions of F[G; η, σ]. The methods of N. I. Dubrovin allowed to show that specific classes of groups can be embedded into a skew field. For example, N. I. Dubrovin contrived some special criteria, which are applicable on the universal covering group of SL(2, R). These methods have also been explored by J. Gr{\"a}ter and R. P. Sperner (cf. [10]) as well as N.H. Halimi and T. Ito (cf. [11]). Furthermore, it is of interest to know if skew fields of fractions are unique. For example, left and right Ore domains have unique skew fields of fractions (cf. [2]). This is not the general case as for example the free group with 2 generators can be embedded into non-isomorphic skew fields of fractions (cf. [12]). It seems likely that Ore domains are the most general case for which unique skew fields of fractions exist. One approach to gain uniqueness is to restrict the search to skew fields of fractions with additional properties. I. Hughes has defined skew fields of fractions of crossed product rings F[G; η, σ] with locally indicable G which fulfill a special condition. These are called Hughes-free skew fields of fractions and I. Hughes has proven that they are unique if they exist [13, 14]. This thesis will connect the ideas of N. I. Dubrovin and I. Hughes. The first chapter contains the basic terminology and concepts used in this thesis. We present methods provided by N. I. Dubrovin such as the complexity of elements in rational closures and special properties of endomorphisms of the vector space of formal power series F((G)). To combine the ideas of N.I. Dubrovin and I. Hughes we introduce Conradian left-ordered groups of maximal rank and examine their connection to locally indicable groups. Furthermore we provide notations for crossed product rings, skew fields of fractions as well as Dubrovin-rings and prove some technical statements which are used in later parts. The second chapter focuses on Hughes-free skew fields of fractions and their connection to Dubrovin-rings. For that purpose we introduce series representations to interpret elements of Hughes-free skew fields of fractions as skew formal Laurent series. This 1 Introduction allows us to prove that for Conradian left-ordered groups G of maximal rank the statement "F[G; η, σ] has a Hughes-free skew field of fractions" implies "The Dubrovin ring of F [G; η, σ] is a skew field". We will also prove the reverse and apply the results to give a new prove of Theorem 1 in [13]. Furthermore we will show how to extend injective ring homomorphisms of some crossed product rings onto their Hughes-free skew fields of fractions. At last we will be able to answer the open question whether Hughes--free skew fields are strongly Hughes-free (cf. [17, page 53]).}, language = {en} } @misc{BeniniSchenkel2017, author = {Benini, Marco and Schenkel, Alexander}, title = {Quantum field theories on categories fibered in groupoids}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {895}, issn = {1866-8372}, doi = {10.25932/publishup-43154}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-431541}, pages = {48}, year = {2017}, abstract = {We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.}, language = {en} } @article{SharmaHainzlZoelleretal.2020, author = {Sharma, Shubham and Hainzl, Sebastian and Z{\"o}ller, Gert and Holschneider, Matthias}, title = {Is Coulomb stress the best choice for aftershock forecasting?}, series = {Journal of geophysical research : Solid earth}, volume = {125}, journal = {Journal of geophysical research : Solid earth}, number = {9}, publisher = {American Geophysical Union}, address = {Washington}, issn = {2169-9313}, doi = {10.1029/2020JB019553}, pages = {12}, year = {2020}, abstract = {The Coulomb failure stress (CFS) criterion is the most commonly used method for predicting spatial distributions of aftershocks following large earthquakes. However, large uncertainties are always associated with the calculation of Coulomb stress change. The uncertainties mainly arise due to nonunique slip inversions and unknown receiver faults; especially for the latter, results are highly dependent on the choice of the assumed receiver mechanism. Based on binary tests (aftershocks yes/no), recent studies suggest that alternative stress quantities, a distance-slip probabilistic model as well as deep neural network (DNN) approaches, all are superior to CFS with predefined receiver mechanism. To challenge this conclusion, which might have large implications, we use 289 slip inversions from SRCMOD database to calculate more realistic CFS values for a layered half-space and variable receiver mechanisms. We also analyze the effect of the magnitude cutoff, grid size variation, and aftershock duration to verify the use of receiver operating characteristic (ROC) analysis for the ranking of stress metrics. The observations suggest that introducing a layered half-space does not improve the stress maps and ROC curves. However, results significantly improve for larger aftershocks and shorter time periods but without changing the ranking. We also go beyond binary testing and apply alternative statistics to test the ability to estimate aftershock numbers, which confirm that simple stress metrics perform better than the classic Coulomb failure stress calculations and are also better than the distance-slip probabilistic model.}, language = {en} } @misc{KarpuzCevikKoppitzetal.2013, author = {Karpuz, Eylem Guzel and {\c{C}}evik, Ahmet Sinan and Koppitz, J{\"o}rg and Cangul, Ismail Naci}, title = {Some fixed-point results on (generalized) Bruck-Reilly ∗-extensions of monoids}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {942}, issn = {1866-8372}, doi = {10.25932/publishup-43270}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-432701}, pages = {11}, year = {2013}, abstract = {In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of arbitrary monoids to be regular, coregular and strongly π-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal.}, language = {en} } @article{AlSaedyTarchanov2020, author = {Al-Saedy, Ammar Jaffar Muhesin and Tarchanov, Nikolaj Nikolaevič}, title = {A degree theory for Lagrangian boundary value problems}, series = {Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics \& physics}, volume = {13}, journal = {Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics \& physics}, number = {1}, publisher = {Sibirskij Federalʹnyj Universitet}, address = {Krasnojarsk}, issn = {1997-1397}, doi = {10.17516/1997-1397-2020-13-1-5-25}, pages = {5 -- 25}, year = {2020}, abstract = {We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems.}, language = {en} } @article{ClavierGuoPaychaetal.2019, author = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {An algebraic formulation of the locality principle in renormalisation}, series = {European Journal of Mathematics}, volume = {5}, journal = {European Journal of Mathematics}, number = {2}, publisher = {Springer}, address = {Cham}, issn = {2199-675X}, doi = {10.1007/s40879-018-0255-8}, pages = {356 -- 394}, year = {2019}, abstract = {We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs.}, language = {en} } @incollection{ClavierGuoPaychaetal.2020, author = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {Renormalisation and locality}, series = {Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2}, booktitle = {Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2}, publisher = {European Mathematical Society Publishing House}, address = {Z{\"u}rich}, isbn = {978-3-03719-205-4 print}, doi = {10.4171/205}, pages = {85 -- 132}, year = {2020}, language = {en} } @inproceedings{AudinDucourtiouxOuedraogoetal.2017, author = {Audin, Mich{\`e}le and Ducourtioux, Catherine and Ou{\´e}draogo, Fran{\c{c}}oise and Schulz, Ren{\´e} and Delgado, Julio and Ruzhansky, Michael and Lebeau, Gilles}, title = {Integral Fourier operators}, editor = {Paycha, Sylvie}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-413-5}, issn = {2199-4951}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-402657}, pages = {xxi, 229}, year = {2017}, abstract = {This volume of contributions based on lectures delivered at a school on Fourier Integral Operators held in Ouagadougou, Burkina Faso, 14-26 September 2015, provides an introduction to Fourier Integral Operators (FIO) for a readership of Master and PhD students as well as any interested layperson. Considering the wide spectrum of their applications and the richness of the mathematical tools they involve, FIOs lie the cross-road of many a field. This volume offers the necessary background, whether analytic or geometric, to get acquainted with FIOs, complemented by more advanced material presenting various aspects of active research in that area.}, language = {en} } @article{ChelkhLyTarkhanov2020, author = {Chelkh, W. and Ly, Ibrahim and Tarkhanov, Nikolai}, title = {A remark on the Laplace transform}, series = {Siberian Mathematical Journal}, volume = {61}, journal = {Siberian Mathematical Journal}, number = {4}, publisher = {Consultants Bureau, Springer}, address = {New York}, issn = {0037-4466}, doi = {10.1134/S0037446620040151}, pages = {755 -- 762}, year = {2020}, abstract = {The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces.}, language = {en} } @article{KayaFreitag2022, author = {Kaya, Adem and Freitag, Melina A.}, title = {Conditioning analysis for discrete Helmholtz problems}, series = {Computers and mathematics with applications : an international journal}, volume = {118}, journal = {Computers and mathematics with applications : an international journal}, publisher = {Elsevier Science}, address = {Amsterdam}, issn = {0898-1221}, doi = {10.1016/j.camwa.2022.05.016}, pages = {171 -- 182}, year = {2022}, abstract = {In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.}, language = {en} } @article{KellerSchwarz2020, author = {Keller, Matthias and Schwarz, Michael}, title = {Courant's nodal domain theorem for positivity preserving forms}, series = {Journal of spectral theory}, volume = {10}, journal = {Journal of spectral theory}, number = {1}, publisher = {EMS Publishing House}, address = {Z{\"u}rich}, issn = {1664-039X}, doi = {10.4171/JST/292}, pages = {271 -- 309}, year = {2020}, abstract = {We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.}, language = {en} } @article{KolasinskiMenne2017, author = {Kolasinski, Slawomir and Menne, Ulrich}, title = {Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds}, series = {Nonlinear Differential Equations and Applications NoDEA}, volume = {24}, journal = {Nonlinear Differential Equations and Applications NoDEA}, publisher = {Springer}, address = {Basel}, issn = {1021-9722}, doi = {10.1007/s00030-017-0436-z}, pages = {56}, year = {2017}, abstract = {This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.}, language = {en} } @article{LyTarkhanov2020, author = {Ly, Ibrahim and Tarkhanov, Nikolaj Nikolaevič}, title = {Asymptotic expansions at nonsymmetric cuspidal points}, series = {Mathematical notes}, volume = {108}, journal = {Mathematical notes}, number = {1-2}, publisher = {Springer Science}, address = {New York}, issn = {0001-4346}, doi = {10.1134/S0001434620070238}, pages = {219 -- 228}, year = {2020}, abstract = {We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.}, language = {en} } @article{Clavier2020, author = {Clavier, Pierre J.}, title = {Double shuffle relations for arborified zeta values}, series = {Journal of algebra}, volume = {543}, journal = {Journal of algebra}, publisher = {Elsevier}, address = {San Diego}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2019.10.015}, pages = {111 -- 155}, year = {2020}, abstract = {Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees.}, language = {en} } @misc{SerthPodlesnyBornsteinetal.2017, author = {Serth, Sebastian and Podlesny, Nikolai and Bornstein, Marvin and Lindemann, Jan and Latt, Johanna and Selke, Jan and Schlosser, Rainer and Boissier, Martin and Uflacker, Matthias}, title = {An interactive platform to simulate dynamic pricing competition on online marketplaces}, series = {2017 IEEE 21st International Enterprise Distributed Object Computing Conference (EDOC)}, journal = {2017 IEEE 21st International Enterprise Distributed Object Computing Conference (EDOC)}, publisher = {Institute of Electrical and Electronics Engineers}, address = {New York}, isbn = {978-1-5090-3045-3}, issn = {2325-6354}, doi = {10.1109/EDOC.2017.17}, pages = {61 -- 66}, year = {2017}, abstract = {E-commerce marketplaces are highly dynamic with constant competition. While this competition is challenging for many merchants, it also provides plenty of opportunities, e.g., by allowing them to automatically adjust prices in order to react to changing market situations. For practitioners however, testing automated pricing strategies is time-consuming and potentially hazardously when done in production. Researchers, on the other side, struggle to study how pricing strategies interact under heavy competition. As a consequence, we built an open continuous time framework to simulate dynamic pricing competition called Price Wars. The microservice-based architecture provides a scalable platform for large competitions with dozens of merchants and a large random stream of consumers. Our platform stores each event in a distributed log. This allows to provide different performance measures enabling users to compare profit and revenue of various repricing strategies in real-time. For researchers, price trajectories are shown which ease evaluating mutual price reactions of competing strategies. Furthermore, merchants can access historical marketplace data and apply machine learning. By providing a set of customizable, artificial merchants, users can easily simulate both simple rule-based strategies as well as sophisticated data-driven strategies using demand learning to optimize their pricing strategies.}, language = {en} } @article{TaghvaeideWiljesMehtaetal.2017, author = {Taghvaei, Amirhossein and de Wiljes, Jana and Mehta, Prashant G. and Reich, Sebastian}, title = {Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem}, series = {Journal of dynamic systems measurement and control}, volume = {140}, journal = {Journal of dynamic systems measurement and control}, number = {3}, publisher = {ASME}, address = {New York}, issn = {0022-0434}, doi = {10.1115/1.4037780}, pages = {11}, year = {2017}, abstract = {This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman-Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.}, language = {en} } @article{deWiljesReichStannat2018, author = {de Wiljes, Jana and Reich, Sebastian and Stannat, Wilhelm}, title = {Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise}, series = {SIAM Journal on Applied Dynamical Systems}, volume = {17}, journal = {SIAM Journal on Applied Dynamical Systems}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1536-0040}, doi = {10.1137/17M1119056}, pages = {1152 -- 1181}, year = {2018}, abstract = {The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.}, language = {en} } @article{HammPelivanGrottetal.2020, author = {Hamm, Maximilian and Pelivan, Ivanka and Grott, Matthias and de Wiljes, Jana}, title = {Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation}, series = {Monthly notices of the Royal Astronomical Society}, volume = {496}, journal = {Monthly notices of the Royal Astronomical Society}, number = {3}, publisher = {Oxford Univ. Press}, address = {Oxford}, issn = {0035-8711}, doi = {10.1093/mnras/staa1755}, pages = {2776 -- 2785}, year = {2020}, abstract = {Deriving thermophysical properties such as thermal inertia from thermal infrared observations provides useful insights into the structure of the surface material on planetary bodies. The estimation of these properties is usually done by fitting temperature variations calculated by thermophysical models to infrared observations. For multiple free model parameters, traditional methods such as least-squares fitting or Markov chain Monte Carlo methods become computationally too expensive. Consequently, the simultaneous estimation of several thermophysical parameters, together with their corresponding uncertainties and correlations, is often not computationally feasible and the analysis is usually reduced to fitting one or two parameters. Data assimilation (DA) methods have been shown to be robust while sufficiently accurate and computationally affordable even for a large number of parameters. This paper will introduce a standard sequential DA method, the ensemble square root filter, for thermophysical modelling of asteroid surfaces. This method is used to re-analyse infrared observations of the MARA instrument, which measured the diurnal temperature variation of a single boulder on the surface of near-Earth asteroid (162173) Ryugu. The thermal inertia is estimated to be 295 +/- 18 Jm(-2) K-1 s(-1/2), while all five free parameters of the initial analysis are varied and estimated simultaneously. Based on this thermal inertia estimate the thermal conductivity of the boulder is estimated to be between 0.07 and 0.12,Wm(-1) K-1 and the porosity to be between 0.30 and 0.52. For the first time in thermophysical parameter derivation, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure that is more than 5000 times more efficient than a comparable parameter sweep.}, language = {en} } @article{AcevedoDeWiljesReich2017, author = {Acevedo, Walter and De Wiljes, Jana and Reich, Sebastian}, title = {Second-order accurate ensemble transform particle filters}, series = {SIAM journal on scientific computing}, volume = {39}, journal = {SIAM journal on scientific computing}, number = {5}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1064-8275}, doi = {10.1137/16M1095184}, pages = {A1834 -- A1850}, year = {2017}, abstract = {Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM T. Sci. Comput., 35, (2013), pp. A2013-A2014[ replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.}, language = {en} } @phdthesis{Rafler2009, author = {Rafler, Mathias}, title = {Gaussian loop- and P{\´o}lya processes : a point process approach}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-029-8}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-38706}, school = {Universit{\"a}t Potsdam}, pages = {ix, 162}, year = {2009}, abstract = {This thesis considers on the one hand the construction of point processes via conditional intensities, motivated by the partial Integration of the Campbell measure of a point process. Under certain assumptions on the intensity the existence of such a point process is shown. A fundamental example turns out to be the P{\´o}lya sum process, whose conditional intensity is a generalisation of the P{\´o}lya urn dynamics. A Cox process representation for that point process is shown. A further process considered is a Poisson process of Gaussian loops, which represents a noninteracting particle system derived from the discussion of indistinguishable particles. Both processes are used to define particle systems locally, for which thermodynamic limits are determined.}, language = {en} } @unpublished{Roelly2013, author = {Roelly, Sylvie}, title = {Reciprocal processes : a stochastic analysis approach}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64588}, year = {2013}, abstract = {Reciprocal processes, whose concept can be traced back to E. Schr{\"o}dinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. L{\´e}onard.}, language = {en} } @unpublished{MeleardRoelly2013, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {Evolutive two-level population process and large population approximations}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64604}, year = {2013}, abstract = {We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.}, language = {en} } @phdthesis{Ludewig2016, author = {Ludewig, Matthias}, title = {Path integrals on manifolds with boundary and their asymptotic expansions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94387}, school = {Universit{\"a}t Potsdam}, pages = {146}, year = {2016}, abstract = {It is "scientific folklore" coming from physical heuristics that solutions to the heat equation on a Riemannian manifold can be represented by a path integral. However, the problem with such path integrals is that they are notoriously ill-defined. One way to make them rigorous (which is often applied in physics) is finite-dimensional approximation, or time-slicing approximation: Given a fine partition of the time interval into small subintervals, one restricts the integration domain to paths that are geodesic on each subinterval of the partition. These finite-dimensional integrals are well-defined, and the (infinite-dimensional) path integral then is defined as the limit of these (suitably normalized) integrals, as the mesh of the partition tends to zero. In this thesis, we show that indeed, solutions to the heat equation on a general compact Riemannian manifold with boundary are given by such time-slicing path integrals. Here we consider the heat equation for general Laplace type operators, acting on sections of a vector bundle. We also obtain similar results for the heat kernel, although in this case, one has to restrict to metrics satisfying a certain smoothness condition at the boundary. One of the most important manipulations one would like to do with path integrals is taking their asymptotic expansions; in the case of the heat kernel, this is the short time asymptotic expansion. In order to use time-slicing approximation here, one needs the approximation to be uniform in the time parameter. We show that this is possible by giving strong error estimates. Finally, we apply these results to obtain short time asymptotic expansions of the heat kernel also in degenerate cases (i.e. at the cut locus). Furthermore, our results allow to relate the asymptotic expansion of the heat kernel to a formal asymptotic expansion of the infinite-dimensional path integral, which gives relations between geometric quantities on the manifold and on the loop space. In particular, we show that the lowest order term in the asymptotic expansion of the heat kernel is essentially given by the Fredholm determinant of the Hessian of the energy functional. We also investigate how this relates to the zeta-regularized determinant of the Jacobi operator along minimizing geodesics.}, language = {en} } @misc{Scharrer2016, type = {Master Thesis}, author = {Scharrer, Christian}, title = {Relating diameter and mean curvature for varifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-97013}, school = {Universit{\"a}t Potsdam}, pages = {42}, year = {2016}, abstract = {The main results of this thesis are formulated in a class of surfaces (varifolds) generalizing closed and connected smooth submanifolds of Euclidean space which allows singularities. Given an indecomposable varifold with dimension at least two in some Euclidean space such that the first variation is locally bounded, the total variation is absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional Hausdorff measure. Moreover, if the support of the weight measure is compact, then the intrinsic diameter with respect to the support of the weight measure is estimated in terms of the generalized mean curvature. This estimate is in analogy to the diameter control for closed connected manifolds smoothly immersed in some Euclidean space of Peter Topping. Previously, it was not known whether the hypothesis in this thesis implies that two points in the support of the weight measure have finite geodesic distance.}, language = {en} } @unpublished{ShlapunovTarkhanov2016, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {An open mapping theorem for the Navier-Stokes equations}, volume = {5}, number = {10}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-98687}, pages = {80}, year = {2016}, abstract = {We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed H{\"o}lder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fr{\´e}chet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of H{\"o}lder spaces.}, language = {en} } @unpublished{BlanchardKraemer2016, author = {Blanchard, Gilles and Kr{\"a}mer, Nicole}, title = {Convergence rates of kernel conjugate gradient for random design regression}, volume = {5}, number = {8}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94195}, pages = {31}, year = {2016}, abstract = {We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.}, language = {en} } @phdthesis{Beinrucker2015, author = {Beinrucker, Andre}, title = {Variable selection in high dimensional data analysis with applications}, school = {Universit{\"a}t Potsdam}, pages = {VII, 107}, year = {2015}, language = {en} } @unpublished{FedosovTarkhanov2015, author = {Fedosov, Boris and Tarkhanov, Nikolai Nikolaevich}, title = {Deformation quantisation and boundary value problems}, volume = {4}, number = {5}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-77150}, pages = {27}, year = {2015}, abstract = {We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.}, language = {en} } @unpublished{ElinShoikhetTarkhanov2015, author = {Elin, Mark and Shoikhet, David and Tarkhanov, Nikolai Nikolaevich}, title = {Analytic semigroups of holomorphic mappings and composition operators}, volume = {4}, number = {6}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-77914}, pages = {30}, year = {2015}, abstract = {In this paper we study the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. We also provide a brief review around this topic.}, language = {en} } @incollection{Petsche2015, author = {Petsche, Hans-Joachim}, title = {Einf{\"u}hrung}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, booktitle = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {9 -- 14}, year = {2015}, language = {de} } @article{Petsche2015, author = {Petsche, Hans-Joachim}, title = {Raum und Zahl - philosophische Kontexte}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {15 -- 33}, year = {2015}, language = {de} } @article{SixtusFischer2015, author = {Sixtus, Elena and Fischer, Martin H.}, title = {Eine kognitionswissenschaftliche Betrachtung der Konzepte "Raum" und "Zahl"}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {35 -- 62}, year = {2015}, language = {de} } @article{Proeve2015, author = {Pr{\"o}ve, Ralf}, title = {Numerische Zeichen und die Repr{\"a}sentation von Sinn}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {63 -- 76}, year = {2015}, language = {de} } @article{Heimann2015, author = {Heimann, Heinz-Dieter}, title = {Kreise - Punkte - Linien}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {111 -- 127}, year = {2015}, language = {de} } @phdthesis{Chutsagulprom2016, author = {Chutsagulprom, Nawinda}, title = {Ensemble-based filters dealing with non-Gaussianity and nonlinearity}, school = {Universit{\"a}t Potsdam}, pages = {95}, year = {2016}, language = {en} } @unpublished{RoellyVallois2016, author = {Roelly, Sylvie and Vallois, Pierre}, title = {Convoluted Brownian motion}, volume = {5}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-96339}, pages = {37}, year = {2016}, abstract = {In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are neither Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the higher-dimensional monomial convoluted Brownian motion.}, language = {en} } @unpublished{FedchenkoTarkhanov2017, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {A Rad{\´o} Theorem for the Porous Medium Equation}, series = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, volume = {6}, journal = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102735}, pages = {12}, year = {2017}, abstract = {We prove that each locally Lipschitz continuous function satisfying the porous medium equation away from the set of its zeroes is actually a weak solution of this equation in the whole domain.}, language = {en} } @unpublished{ShlapunovTarkhanov2017, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Golusin-Krylov Formulas in Complex Analysis}, series = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, volume = {6}, journal = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, number = {2}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102774}, pages = {25}, year = {2017}, abstract = {This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.}, language = {en} } @phdthesis{Pirhayati2016, author = {Pirhayati, Mohammad}, title = {Edge operators and boundary value problems}, school = {Universit{\"a}t Potsdam}, pages = {VI, 119}, year = {2016}, language = {en} } @phdthesis{Lyu2016, author = {Lyu, Xiaojing}, title = {Operators on singular manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-103643}, school = {Universit{\"a}t Potsdam}, pages = {117}, year = {2016}, abstract = {We study the interplay between analysis on manifolds with singularities and complex analysis and develop new structures of operators based on the Mellin transform and tools for iterating the calculus for higher singularities. We refer to the idea of interpreting boundary value problems (BVPs) in terms of pseudo-differential operators with a principal symbolic hierarchy, taking into account that BVPs are a source of cone and edge operator algebras. The respective cone and edge pseudo-differential algebras in turn are the starting point of higher corner theories. In addition there are deep relationships between corner operators and complex analysis. This will be illustrated by the Mellin symbolic calculus.}, language = {en} } @phdthesis{Gopalakrishnan2016, author = {Gopalakrishnan, Sathej}, title = {Mathematical modelling of host-disease-drug interactions in HIV disease}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100100}, school = {Universit{\"a}t Potsdam}, pages = {121}, year = {2016}, abstract = {The human immunodeficiency virus (HIV) has resisted nearly three decades of efforts targeting a cure. Sustained suppression of the virus has remained a challenge, mainly due to the remarkable evolutionary adaptation that the virus exhibits by the accumulation of drug-resistant mutations in its genome. Current therapeutic strategies aim at achieving and maintaining a low viral burden and typically involve multiple drugs. The choice of optimal combinations of these drugs is crucial, particularly in the background of treatment failure having occurred previously with certain other drugs. An understanding of the dynamics of viral mutant genotypes aids in the assessment of treatment failure with a certain drug combination, and exploring potential salvage treatment regimens. Mathematical models of viral dynamics have proved invaluable in understanding the viral life cycle and the impact of antiretroviral drugs. However, such models typically use simplified and coarse-grained mutation schemes, that curbs the extent of their application to drug-specific clinical mutation data, in order to assess potential next-line therapies. Statistical models of mutation accumulation have served well in dissecting mechanisms of resistance evolution by reconstructing mutation pathways under different drug-environments. While these models perform well in predicting treatment outcomes by statistical learning, they do not incorporate drug effect mechanistically. Additionally, due to an inherent lack of temporal features in such models, they are less informative on aspects such as predicting mutational abundance at treatment failure. This limits their application in analyzing the pharmacology of antiretroviral drugs, in particular, time-dependent characteristics of HIV therapy such as pharmacokinetics and pharmacodynamics, and also in understanding the impact of drug efficacy on mutation dynamics. In this thesis, we develop an integrated model of in vivo viral dynamics incorporating drug-specific mutation schemes learned from clinical data. Our combined modelling approach enables us to study the dynamics of different mutant genotypes and assess mutational abundance at virological failure. As an application of our model, we estimate in vivo fitness characteristics of viral mutants under different drug environments. Our approach also extends naturally to multiple-drug therapies. Further, we demonstrate the versatility of our model by showing how it can be modified to incorporate recently elucidated mechanisms of drug action including molecules that target host factors. Additionally, we address another important aspect in the clinical management of HIV disease, namely drug pharmacokinetics. It is clear that time-dependent changes in in vivo drug concentration could have an impact on the antiviral effect, and also influence decisions on dosing intervals. We present a framework that provides an integrated understanding of key characteristics of multiple-dosing regimens including drug accumulation ratios and half-lifes, and then explore the impact of drug pharmacokinetics on viral suppression. Finally, parameter identifiability in such nonlinear models of viral dynamics is always a concern, and we investigate techniques that alleviate this issue in our setting.}, language = {en} } @unpublished{VasilievTarkhanov2016, author = {Vasiliev, Serguei and Tarkhanov, Nikolai Nikolaevich}, title = {Construction of series of perfect lattices by layer superposition}, volume = {5}, number = {11}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100591}, pages = {11}, year = {2016}, abstract = {We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.}, language = {en} } @phdthesis{Bartels1999, author = {Bartels, Knut}, title = {Tests zur Modellspezifikation in der nichtlinearen Regression}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0000171}, school = {Universit{\"a}t Potsdam}, year = {1999}, abstract = {Als Grundlage vieler statistischer Verfahren wird der Prozess der Entstehung von Daten modelliert, um dann weitere Sch{\"a}tz- und Testverfahren anzuwenden. Diese Arbeit befasst sich mit der Frage, wie diese Spezifikation f{\"u}r parametrische Modelle selbst getestet werden kann. In Erweiterung bestehender Verfahren werden Tests mit festem Kern eingef{\"u}hrt und ihre asymptotischen Eigenschaften werden analysiert. Es wird gezeigt, dass die Bestimmung der kritischen Werte mit mehreren Stichprobenwiederholungsverfahren m{\"o}glich ist. Von diesen ist eine neue Monte-Carlo-Approximation besonders wichtig, da sie die Komplexit{\"a}t der Berechnung deutlich verringern kann. Ein bedingter Kleinste-Quadrate-Sch{\"a}tzer f{\"u}r nichtlineare parametrische Modelle wird definiert und seine wesentlichen asymptotischen Eigenschaften werden hergeleitet. S{\"a}mtliche Versionen der Tests und alle neuen Konzepte wurden in Simulationsstudien untersucht, deren wichtigste Resultate pr{\"a}sentiert werden. Die praktische Anwendbarkeit der Testverfahren wird an einem Datensatz zur Produktwahl dargelegt, der mit multinomialen Logit-Modellen analysiert werden soll.}, language = {de} } @article{Meyerhoefer2001, author = {Meyerh{\"o}fer, Wolfram}, title = {Was misst TIMSS?}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-224}, pages = {1 -- 11}, year = {2001}, abstract = {Bei der Erstellung und Interpretation mathematischer Leistungstests steht die Frage, was eine Aufgabe mißt. Der Artikel stellt mit der strukturalen oder objektiven Hermeneutik eine Methode vor, mit der die verschiedenen Dimensionen der von einer Aufgabe erfassten F{\"a}higkeiten herausgearbeitet werden k{\"o}nnen. Dabei werden fachliche Anforderungen, Irritationsmomente und das durch die Aufgabe transportierte Bild vom jeweiligen Fach ebenso erfasst wie Momente, die man eher als Testf{\"a}higkeit bezeichnen w{\"u}rde.Am Beispiel einer TIMSS-Aufgabe wird diskutiert, dass das von den Testerstellern benutzte theoretische Konstrukt kaum geeignet ist, nachhaltig zu beschreiben, was eine Aufgabe misst.}, language = {de} } @misc{DaiPraLouisMinelli2006, author = {Dai Pra, Paolo and Louis, Pierre-Yves and Minelli, Ida}, title = {Monotonicity and complete monotonicity for continuous-time Markov chains}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-7665}, year = {2006}, abstract = {We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.}, subject = {Stochastik}, language = {en} } @unpublished{RoellyFradon2006, author = {Roelly, Sylvie and Fradon, Myriam}, title = {Infinite system of Brownian balls : equilibrium measures are canonical Gibbs}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6720}, year = {2006}, abstract = {We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.}, language = {en} } @phdthesis{Rosenberger2006, author = {Rosenberger, Elke}, title = {Asymptotic spectral analysis and tunnelling for a class of difference operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-7393}, school = {Universit{\"a}t Potsdam}, year = {2006}, abstract = {We analyze the asymptotic behavior in the limit epsilon to zero for a wide class of difference operators H_epsilon = T_epsilon + V_epsilon with underlying multi-well potential. They act on the square summable functions on the lattice (epsilon Z)^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.}, subject = {Mathematische Physik}, language = {en} } @unpublished{MeklerBoeckmannSokolovskaia2000, author = {Mekler, A. A. and B{\"o}ckmann, Christine and Sokolovskaia, N.}, title = {Particle distribution from spectral Mie-scattering: kernel representation and singular-value spectrum}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14948}, year = {2000}, abstract = {This paper deals with the Mie scattering kernels for multi-spectral data. The kernels may be represented in form of power series. Furthermore, the singular-value spectrum and the degree of ill-posedness in dependence on the refractive index of the particles are numerically approximated. A special hybrid regularization technique allows us to determine via inversion the particle distributions of different types.}, language = {en} } @unpublished{KapanadzeSchulze2000, author = {Kapanadze, David and Schulze, Bert-Wolfgang}, title = {Boundary value problems on manifolds with exits to infinity}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25727}, year = {2000}, abstract = {We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical exits to infinity. The symbols are classical both in covariables and variables. The operators are determined by principal symbol tuples modulo operators of lower orders and weights (such remainders are compact in weighted Sobolev spaces). We develop the concept of ellipticity, construct parametrices within the algebra and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the Atiyah-Bott condition.}, language = {en} } @unpublished{SavinSternin2000, author = {Savin, Anton and Sternin, Boris}, title = {Eta-invariant and Pontrjagin duality in K-theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25747}, year = {2000}, abstract = {The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.}, language = {en} } @unpublished{SchulzeTarkhanov2000, author = {Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {Pseudodifferential operators on manifolds with corners}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25783}, year = {2000}, abstract = {We describe an algebra of pseudodifferential operators on a manifold with corners.}, language = {en} } @unpublished{SchulzeSterninShatalov1997, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {On the index of differential operators on manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-24965}, year = {1997}, abstract = {The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.}, language = {en} } @unpublished{AirapetyanWitt1997, author = {Airapetyan, Ruben and Witt, Ingo}, title = {Isometric properties of the Hankel Transformation in weighted sobolev spaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25001}, year = {1997}, abstract = {It is shown that the Hankel transformation Hsub(v) acts in a class of weighted Sobolev spaces. Especially, the isometric mapping property of Hsub(v) which holds on L²(IRsub(+),rdr) is extended to spaces of arbitrary Sobolev order. The novelty in the approach consists in using techniques developed by B.-W. Schulze and others to treat the half-line Rsub(+) as a manifold with a conical singularity at r = 0. This is achieved by pointing out a connection between the Hankel transformation and the Mellin transformation.The procedure proposed leads at the same time to a short proof of the Hankel inversion formula. An application to the existence and higher regularity of solutions, including their asymptotics, to the 1-1-dimensional edge-degenerated wave equation is given.}, language = {en} } @unpublished{Korkey1998, author = {Korkey, Michael Brian}, title = {Optimal factorization of Muckenhoupt weights}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25266}, year = {1998}, abstract = {Peter Jones' theorem on the factorization of Ap weights is sharpened for weights with bounds near 1, allowing the factorization to be performed continuously near the limiting, unweighted case. When 1 < p < infinite and omega is an Ap weight with bound Ap(omega) = 1 + epsilon, it is shown that there exist Asub1 weights u, v such that both the formula omega = uv(1-p) and the estimates A1 (u), A1 (v) = 1 + Omikron (√epsilon) hold. The square root in these estimates is also proven to be the correct asymptotic power as epsilon -> 0.}, language = {en} } @unpublished{SchulzeNazaikinskiiSternin1998, author = {Schulze, Bert-Wolfgang and Nazaikinskii, Vladimir and Sternin, Boris}, title = {The index of quantized contact transformations on manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25276}, year = {1998}, abstract = {The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator.}, language = {en} } @unpublished{SavinSchulzeSternin1998, author = {Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {On the invariant index formulas for spectral boundary value problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25285}, year = {1998}, abstract = {In the paper we study the possibility to represent the index formula for spectral boundary value problems as a sum of two terms, the first one being homotopy invariant of the principal symbol, while the second depends on the conormal symbol of the problem only. The answer is given in analytical, as well as in topological terms.}, language = {en} } @unpublished{SchulzeNazaikinskiiSternin1998, author = {Schulze, Bert-Wolfgang and Nazaikinskii, Vladimir and Sternin, Boris}, title = {A semiclassical quantization on manifolds with singularities and the Lefschetz Formula for Elliptic Operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25296}, year = {1998}, abstract = {For general endomorphisms of elliptic complexes on manifolds with conical singularities, the semiclassical asymptotics of the Atiyah-Bott-Lefschetz number is calculated in terms of fixed points of the corresponding canonical transformation of the symplectic space.}, language = {en} } @unpublished{ChenHidetoshi1998, author = {Chen, Hua and Hidetoshi, Tahara}, title = {On the holomorphic solution of non-linear totally characteristic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25333}, year = {1998}, abstract = {The paper deals with a non-linear singular partial differential equation: (E) t∂/∂t = F(t, x, u, ∂u/∂x) in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by G{\´e}rard-Tahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case.}, language = {en} }